Risk-Sensitive Investment Management via Free Energy-Entropy Duality

TL;DR

This paper introduces a novel approach to risk-sensitive portfolio optimization using free energy-entropy duality, providing explicit solutions and interpretations, and demonstrating its practical implementation with reinforcement learning on U.S. equity data.

Contribution

It reformulates the benchmarked risk-sensitive problem as a linear-quadratic-Gaussian game, linking classical methods with modern duality and RL techniques.

Findings

Explicit affine feedback controls derived for the portfolio problem.

The optimal allocation can be viewed as a fractional Kelly strategy.

Numerical experiments confirm the framework's tractability and equivalence of approaches.

Abstract

We study a benchmarked risk-sensitive portfolio problem in a factor-based setting to bring together three strands of the literature: benchmarked risk-sensitive investment management, the Kuroda-Nagai change-of-measure method, and the free energy-entropy duality of Dai Pra et al. (1996). We show that the duality yields a direct solution of the benchmarked problem by reformulating it as a linear-quadratic-Gaussian stochastic differential game under a suitable equivalent probability measure, with an entropic regularization. The resulting value function is quadratic, the optimal controls are explicit affine feedback maps, and the optimal allocation admits two complementary interpretations: as a fractional Kelly strategy and as a Kelly portfolio adjusted via the entropic regularization. This formulation, therefore, contributes both a direct analytical route to the solution and a clearer interpretation of risk sensitivity, thereby embedding the classical Kuroda-Nagai change-of-measure approach within a more general framework. An added benefit of this formulation is that it is suitable for implementation via an RL algorithm. A simple implementation on U.S. equity data illustrates the tractability of the framework and numerically confirms the equivalence of the two approaches.